PHYSICS 201 (FALL 2009)
MIDTERM EXAM 2
Given: November 20,4:00 pm
Due: November 21, 12:00 noon
Dear students:
To let you have the benefit of all possible aids at your disposal, this exam is a take-home
one. But please note that, while you may consult any number of resources in print, you
CANNOT have consultation with ANY other person ----- DEAD OR ALIVE!
1. (i) Using the calculus of residues, evaluate the sum.
f
(n 2--+ c,.:.
2- )
hl. -+-
(
J/)
(ii) Now, let a and b go to zero to show that
00
2. Determine the limiting value of the function/(n),
-n
n
f
(r)
if)
defined by the product
1
2
...
• 2
,
'fJ -I
,
J =/
..
lfYl
_----)
ljn2-_1
as n tends to infinity.
[Hint: Express/(n)
in terms of gamma/factorial functions, and use Stirling's formula].
3. Apply Laplace's method to show that, for large
J
DO
L
6- f-.!.. J,. t22-.
Y,
.jT
t VAt z
o
To be more specific, the derivation here assumes that (b
.-!-,
iv.
Y/
a2) » 1.
/
4. Using the method of stationary phase, determine the'asymptotic behavior of the
function
for x » 1.
5. Using the method of steepest descents, show that, for large n,
Here, n may be regarded as running through integral values only (so that you don't have
to employ a "branch cut").
.
Also note that, of the two saddle points you have in this problem, the contour C can be
steered ONLY through the one in the upper-half plane --- NOT through the ony in the
lower-half plane. Why this is so --- I'll explain in the class!
'
6. Solve the differential equation
/
3y" + 2y' - 8y = 5 cos x
for both the complementary function and the particular solution.
7. (i) Apply the Frobenius method to obtain the series solutions of the differential
equation
(ii) Next, making a substitution such as y (x) = I(x) z (x), put the above equation
into the form
z
Q (x) z
and apply the WKB method to determine the asymptotic behavior of the function z (x),
and hence ofy (x), as x goes to infinity.
-C.
00
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